Book contents
- Frontmatter
- Contents
- Preface
- Abbreviations
- 1 Introduction
- 2 Introduction to spectral methods via orthogonal functions
- 3 Introduction to PS methods via finite differences
- 4 Key properties of PS approximations
- 5 PS variations and enhancements
- 6 PS methods in polar and spherical geometries
- 7 Comparisons of computational cost for FD and PS methods
- 8 Applications for spectral methods
- Appendices
- A Jacobi polynomials
- B Tau, Galerkin, and collocation (PS) implementations
- C Codes for algorithm to find FD weights
- D Lebesgue constants
- E Potential function estimate for polynomial interpolation error
- F FFT-based implementation of PS methods
- G Stability domains for some ODE solvers
- H Energy estimates
- References
- Index
H - Energy estimates
Published online by Cambridge University Press: 03 December 2009
- Frontmatter
- Contents
- Preface
- Abbreviations
- 1 Introduction
- 2 Introduction to spectral methods via orthogonal functions
- 3 Introduction to PS methods via finite differences
- 4 Key properties of PS approximations
- 5 PS variations and enhancements
- 6 PS methods in polar and spherical geometries
- 7 Comparisons of computational cost for FD and PS methods
- 8 Applications for spectral methods
- Appendices
- A Jacobi polynomials
- B Tau, Galerkin, and collocation (PS) implementations
- C Codes for algorithm to find FD weights
- D Lebesgue constants
- E Potential function estimate for polynomial interpolation error
- F FFT-based implementation of PS methods
- G Stability domains for some ODE solvers
- H Energy estimates
- References
- Index
Summary
Different variations of the “energy method” can be used to show that PDEs are well posed, to show that discrete approximations are stable, and to establish (global) convergence rates under mesh refinements. The energy approach is very broadly applicable, and can handle many cases that include: boundary conditions; variable coefficients (and nonlinearities); and nonperiodic PS methods.
However, this flexibility and power comes at a price of often significant technical difficulty. This appendix is intended to provide only a first flavor of this rich subject to readers who are unfamiliar with it. For this purpose, we here consider five examples, all relating to the heat equation on the interval [–1, 1]. More systematic descriptions can be found in Richtmyer and Morton (1967), Gustafsson et al. (1995), and (for PS methods) Canuto et al. (1988).
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- Information
- A Practical Guide to Pseudospectral Methods , pp. 210 - 216Publisher: Cambridge University PressPrint publication year: 1996